Solve for $x$ : $2x^2 - 24x + 54 = 0$
Explanation: Dividing both sides by $2$ gives: $ x^2 {-12}x + {27} = 0 $ The coefficient on the $x$ term is $-12$ and the constant term is $27$ , so we need to find two numbers that add up to $-12$ and multiply to $27$ The two numbers $-3$ and $-9$ satisfy both conditions: $ {-3} + {-9} = {-12} $ $ {-3} \times {-9} = {27} $ $(x {-3}) (x {-9}) = 0$ Since the following equation is true we know that one or both quantities must equal zero. $(x -3) (x -9) = 0$ $x - 3 = 0$ or $x - 9 = 0$ Thus, $x = 3$ and $x = 9$ are the solutions.